An empirical method for comparing the shape of two Gaussian mixtures

The motivation of this study is to be able to recognize planar objects consisting of “blobs” which can be modeled as Gaussian mixtures densities. Given are a two planar point-sets P̂ and P̃ consisting of point samples drawn from Gaussian mixtures ρ̂(x) and ρ̃(x), respectively. We propose a method to determine whether ρ̂(x) and ρ̃(x) have the same shape using P̂ and P̃. More precisely, we empirically compare the underlying distribution of distances of ρ̂(x) and ρ̃(x) using pairwise distances of the points contained in P̂ and P̃, respectively. The distribution of distances has been shown to be a lossless representation of generic Gaussian mixtures. Since distances are invariant under rotations and translations, this provides a workaround to the problem of aligning the objects before comparing them. We assess the method using synthetic data as well as real data consisting of halftoning patterns. Our results show a robust recognition performance.